The Alternative

A theorem of the alternative asserts that either of two conditions holds, but not both simultaneously. Theorems of the alternative have played a major role in the development of mathematical economics. See a theorem of the alternative as an existence theorem: the failure of one condition to hold implies the other condition. This chapter develops the concept of the alternative via perturbation duality. The primal/dual pair has no duality gap. The optimum value is zero if the first alternative holds and the second does not, and is infinity if the second alternative holds and the first does not. In the historic development of the alternative, the mathematician posits the alternative out of a hat, without explanation or motivation, and then proves the result. In contrast, the duality approach arrives at the alternative by calculating the dual. In this chapter the starting point is a cone alternative. Express one alternative as a closed, convex set. One derives the second alternative from the polar of the cone representation of this set. An example is the efficiency alternative, which reveals the close linkage between efficiency and weak efficiency. Setting the analysis in a general Euclidean Jordan algebra advances the theory. The classical examples of the alternative involve linear inequalities for which the solution set is polyhedral. Even though this latter property does not hold for a general Euclidean Jordan algebra, the classical theorems nevertheless remain valid.